## SectionB.3Cartesian Products

When $X$ and $Y$ are sets, the cartesian product of $X$ and $Y\text{,}$ denoted $X\times Y$, is defined by

\begin{equation*} X\times Y=\{(x,y): x\in X \text{ and } y\in Y\} \end{equation*}

For example, if $X=\{a,b\}$ and $Y=[3]\text{,}$ then

\begin{equation*} X\times Y=\{(a,1),(b,1),(a,2),(b,2),(a,3),(b,3)\}. \end{equation*}

Elements of $X\times Y$ are called ordered pairs. When $p=(x,y)$ is an ordered pair, the element $x$ is referred to as the first coordinate of $p$ while $y$ is the second coordinate of $p\text{.}$ Note that if either $X$ or $Y$ is the empty set, then $X\times Y=\emptyset\text{.}$

###### ExampleB.3.

Let $X=\{\emptyset,(1,0),\{\emptyset\}\}$ and $Y=\{(\emptyset,0)\}\text{.}$ Is $((1,0),\emptyset)$ a member of $X\times Y\text{?}$

Cartesian products can be defined for more than two factors. When $n\ge 2$ is a positive integer and $X_1,X_2,\dots,X_n$ are non-empty sets, their cartesian product is defined by

\begin{equation*} X_1\times X_2\times\dots\times X_n=\{(x_1,x_2,\dots,x_n): x_i\in X_i \text{ for } i = 1,2,\dots,n\} \end{equation*}